DIFFERENTIAL GRADED LIE ALGEBRAS AND FORMAL DEFORMATION THEORY
DIFFERENTIAL GRADED LIE ALGEBRAS AND FORMAL DEFORMATION THEORY
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微分梯度李代数和形式变形理论
DOI：
10.1090/pspum/080.2/2483955
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
M. Manetti
中科院分区：
文献类型：
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作者：
M. Manetti
本文旨在做两件事：（1）给出微分分次李代数、Artin环的函子和障碍物的教程介绍;（2）解释最近一些关于复Kähler流形变形的障碍物消失定理的论文[29，31，32，11，17]的思想和技巧。我们假设读者有一个基本知识的代数几何，同调代数和变形理论;对于这个问题，年轻人可能会阅读优秀的临时文章的阿尔卡塔的诉讼[39]。其共同点是由Quillen，Deligne和Drinfeld提出的以下指导原则：在特征0中，每个变形问题都由微分分次李代数控制。在必要的背景之后，我们将以一种不那么模糊的形式重申这一原则（原则1.9）。指导原则一直局限于抽象概念和个人交流的领域，直到[37，13，24，25]1的出现，在那里，它的巧妙使用已经允许在具体变形问题中的有趣应用。特别是课堂讲稿[24]给出了指导原则（称为那里的元定理）有效性的严肃和令人信服的动机。在本文中，我们应用这些想法，以证明消失定理的障碍空间。为了解释我们研究的主题，考虑一个紧致复流形X的变形的例子，它具有全纯切丛ΘX。著名的Kuranishi定理[26，39，5，14]断言，在复空间Def（X）的芽上存在X的变形X f −→Def（X），其性质是Kodaira-Spencer映射TDef（X）→ H1（X，ΘX）是双射的，并且X在解析芽S上的每个变形同构于f被全纯映射S → Def（X）拉回。从Kuranishi的证明如下，此外：1。Def（X）q−1（0），其中q：H1（X，ΘX）→ H2（X，ΘX）是全纯映射的芽，使得q（0）= 0。2. q在0处的微分是微不足道的。3. q的Mac-Laurin级数的二次部分同构于二次映射H（X，ΘX）→ H（X，θX），x → 1 2 [x，x]，
This paper aims to do two things: (1) to give a tutorial introduction to differential graded Lie algebras, functors of Artin rings and obstructions; (2) to explain ideas and techniques underlying some recent papers [29, 31, 32, 11, 17] concerning vanishing theorems for obstructions to deformations of complex Kähler manifolds. We assume that the reader has a basic knowledge of algebraic geometry, homological algebra and deformation theory; for this topic, the young person may read the excellent expository article of Arcata’s proceedings [39]. The common denominator is the following guiding principle, proposed by Quillen, Deligne and Drinfeld: in characteristic 0 every deformation problem is governed by a differential graded Lie algebra. After the necessary background we will restate such principle in a less vague form (Principle 1.9). The guiding principle has been confined in the realm of abstract ideas and personal communications until the appearance of [37, 13, 24, 25]1 where a clever use of it has permitted interesting applications in concrete deformation problems. In particular the lecture notes [24] give serious and convincing motivations for the validity of the guiding principle (called there meta-theorem). In this paper we apply these ideas in order to prove vanishing theorems for obstruction spaces. Just to explain the subject of our investigation, consider the example of deformations of a compact complex manifold X with holomorphic tangent bundle ΘX . The well known Kuranishi’s theorem [26, 39, 5, 14] asserts that there exists a deformation X f −→Def(X) of X over a germ of complex space Def(X) with the property that the Kodaira-Spencer map TDef(X) → H1(X, ΘX) is bijective and every deformation of X over an analytic germ S is isomorphic to the pull-back of f by a holomorphic map S → Def(X). From Kuranishi’s proof follows moreover that: 1. Def(X) q−1(0), where q : H1(X, ΘX) → H2(X, ΘX) is a germ of holomorphic map such that q(0) = 0. 2. The differential of q at 0 is trivial. 3. The quadratic part of the Mac-Laurin series of q is isomorphic to the quadratic map H(X, ΘX)→ H(X, ΘX), x → 1 2 [x, x],
变形理论、同调代数和镜像对称
DOI：
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发表时间：
2003
期刊：
Ser. High Energy Physics. Cosmol. Gravit.
影响因子：
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作者：
Kaoru Ono;Kenji Fukaya
通讯作者：
Kenji Fukaya